Linear Algebra

What's the big idea of Linear Algebra? **Course Intro**.
What is a Solution to a Linear System? **Intro**.
Visualizing Solutions to Linear Systems - - 2D & 3D Cases Geometrically.
Rewriting a Linear System using Matrix Notation.
Using Elementary Row Operations to Solve Systems of Linear Equations.
Using Elementary Row Operations to simplify a linear system.
Examples with 0, 1, and infinitely many solutions to linear systems.
Row Echelon Form and Reduced Row Echelon Form.
Back Substitution with infinitely many solutions.
What is a vector? Visualizing Vector Addition & Scalar Multiplication.
Introducing Linear Combinations & Span.
How to determine if one vector is in the span of other vectors?.
Matrix-Vector Multiplication and the equation Ax=b.
Matrix-Vector Multiplication Example.
Proving Algebraic Rules in Linear Algebra --- Ex: A(b+c) = Ab +Ac.
The Big Theorem, Part I.
Writing solutions to Ax=b in vector form.
Geometric View on Solutions to Ax=b and Ax=0..
Three nice properties of homogeneous systems of linear equations.
Linear Dependence and Independence - Geometrically.
Determining Linear Independence vs Linear Dependence.
Making a Math Concept Map | Ex: Linear Independence.
Transformations and Matrix Transformations.
Three examples of Matrix Transformations.
Linear Transformations.
Are Matrix Transformations and Linear Transformation the same? Part I.
Every vector is a linear combination of the same n simple vectors!.
Matrix Transformations are the same thing as Linear Transformations.
Finding the Matrix of a Linear Transformation.
One-to-one, Onto, and the Big Theorem Part II.
The motivation and definition of Matrix Multiplication.
Computing matrix multiplication.
Visualizing Composition of Linear Transformations **aka Matrix Multiplication**.
Elementary Matrices.
You can "invert" matrices to solve equations...sometimes!.
Finding inverses to 2x2 matrices is easy!.
Find the Inverse of a Matrix.
When does a matrix fail to be invertible? Also more "Big Theorem"..
Visualizing Invertible Transformations (plus why we need one-to-one).
Invertible Matrices correspond with Invertible Transformations **proof**.
Determinants - a "quick" computation to tell if a matrix is invertible.
Determinants can be computed along any row or column - choose the easiest!.
Vector Spaces | Definition & Examples.
The Vector Space of Polynomials: Span, Linear Independence, and Basis.
Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples.
The Span is a Subspace | Proof + Visualization.
The Null Space & Column Space of a Matrix | Algebraically & Geometrically.
The Basis of a Subspace.
Finding a Basis for the Nullspace or Column space of a matrix A.
Finding a basis for Col(A) when A is not in REF form..
Coordinate Systems From Non-Standard Bases | Definitions + Visualization.
Writing Vectors in a New Coordinate System **Example**.
What Exactly are Grid Lines in Coordinate Systems?.
The Dimension of a Subspace | Definition + First Examples.
Computing Dimension of Null Space & Column Space.
The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!.
Changing Between Two Bases | Derivation + Example.
Visualizing Change Of Basis Dynamically.
Example: Writing a vector in a new basis.
What eigenvalues and eigenvectors mean geometrically.
Using determinants to compute eigenvalues & eigenvectors.
Example: Computing Eigenvalues and Eigenvectors.
A range of possibilities for eigenvalues and eigenvectors.
Diagonal Matrices are Freaking Awesome.
How the Diagonalization Process Works.
Compute large powers of a matrix via diagonalization.
Full Example: Diagonalizing a Matrix.
COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**.
Visualizing Diagonalization & Eigenbases.
Similar matrices have similar properties.
The Similarity Relationship Represents a Change of Basis.
Dot Products and Length.
Distance, Angles, Orthogonality and Pythagoras for vectors.
Orthogonal bases are easy to work with!.
Orthogonal Decomposition Theorem Part 1: Defining the Orthogonal Complement.
The geometric view on orthogonal projections.
Orthogonal Decomposition Theorem Part II.
Proving that orthogonal projections are a form of minimization.
Using Gram-Schmidt to orthogonalize a basis.
Full example: using Gram-Schmidt.
Least Squares Approximations.
Reducing the Least Squares Approximation to solving a system.